Manufacturing engineering production line scheduling management technology integrating availability constraints and heuristic rules

3.1 Construction of PLSM model based on AC and HR

The assembly line problem in the manufacturing industry belongs to a strongly non-deterministic polynomial time hard (NP-hard) problem. Solving this problem usually uses HR-based algorithms [17]. However, the problem faced this time is that in the case of two-stage ALW, as well as the existence of workpiece outsourcing and machine availability constraints in the self-produced workshop, and the second process is the hindering process, its processing time is much longer than the first process. Therefore, to solve scheduling and AC problems, this study combines HR and JR to construct a hybrid algorithm for optimizing scheduling management. This requires first understanding the problem being solved, as described in Figure 1.

Figure 1

Problem description.

In Figure 1, the first is to describe the processing start time of each workpiece in the first working program and describe the processing consumption time in the second working program. Among them, the expression for calculating the completion time of any workpiece in the first work program is given by the following equation:

In Eq. (1), i is the i-th workpiece and ∀ i ≤ n . When the value of j is 1, it represents that the workpiece is being processed in the workshop owned by the enterprise itself. When the value of j is 2, it represents that the workpiece is outsourced by the enterprise to an external workshop for processing, and j = 1 , 2 . S i and F i are the opening and ending times of any workpiece i during the first working program. p i j is the consumption time of the first machining program of workpiece i on the machine in workshop j . p i 1 is the time consumed for processing in the workshop owned by the enterprise itself. p i 2 is the processing time of the first process of workpiece i on the outsourced workshop machine. A variable with x i equal to 0 or 1: When it is equal to 0, it represents that the workpiece i has not been processed in the outsourcing workshop. When it is equal to 1, it represents that the enterprise has entrusted workpiece i to other enterprises for processing. The calculation formula for the time point C i when any workpiece i completes the second work program is shown in the following equation:

In Eq. (2), q i j is the time consumed by workshop j for the second working procedure on workpiece i . q i 2 is the time consumed by the enterprise to entrust workpiece i to an external workshop for the second work procedure. According to Eqs. (1) and (2), the processing speed of workpiece i is not consistent between the outsourcing workshop and the self-produced workshop. Therefore, there may be two situations where the processing speed of workpiece i in the self-produced workshop is greater or less than its processing speed in the outsourced workshop. To make the research more scientific and reasonable, after reviewing relevant literature and the actual working conditions of workpiece processing, it is found that most of the situations that exist in the real manufacturing environment are that the processing speed of self-produced workshops is slightly lower than that of outsourced workshops [18]. Second, when there are other workpieces φ i being processed before the workpiece i is processed in the self-produced workshop or outsourced workshop, the formula can be obtained as follows:

In Eq. (3), S i is the start time of processing workpiece i in the first process. n is the number of workpieces to be processed. S k is the start time of the first processing of workpiece k before workpiece i . When workpiece i is processed in the in-house workshop or outsourced workshop and ranked first, its calculation is represented as F δ 1 = 0 . Among them, δ i refers to the workpiece processed in the same workshop before the processing of workpiece i . If the processing sequence δ i of workpiece i is specified in the contract, the expression is

In Eq. (4), S l is that the start time of workpiece l in the first process is later than that of workpiece k . Based on the above content, a PLSM model based on AC and HR can be constructed, as shown in Figure 2.

Figure 2

Optimization model of two-machine ALW based on outsourcing and ACs.

Figure 2 shows a mathematical programming model that involves objective functions and different element constraints. Conditional constraints include that at a certain point in time, a machine can only process one workpiece, the connection between the first and second processes, the workpiece can only be processed in one workshop, the range of values for each of the two processes, and whether the machine that processes the workpiece in the second process is undergoing maintenance before processing (if it is undergoing maintenance, it needs to wait for the maintenance to be completed before processing). First, the objective function is shown

In Eq. (5), μ represents the transportation cost per batch of the workpiece after it is processed in the outsourcing workshop and transported to the self-produced workshop. σ 1 and σ 2 are the unit production costs of the self-produced workshop and the outsourced workshop. w is the time target parameter. 1 − w is the cost target parameter. The processing cost relationship between the two workshops is shown

In Eq. (6), θ is the standard deviation between the rush cost per unit time spent by the enterprise’s own workshop and the workshop of the commissioned processing enterprise. Among them, m is the workshop production capacity of external enterprises. α represents the total number of outsourced processing tasks. The relationship between m and α is 0 < m < 1 and α < 0 . The relationship between θ and m is shown in Figure 3.

Figure 3

Relationship between θ, m, and the number of artifacts.

In Figure 3, as x i increases, the value of θ tends to be m times that of θ . The larger the value of m , the worse the effect of outsourcing the workpiece, and vice versa. Second, at a certain point in time, the constraint formula that a machine can only process one workpiece is shown

Next, the connection constraints between the two processes and the time constraints for repairing the workpiece in case of a malfunction on the second machine are shown in the following equation:

In Eq. (8), Γ represents the time it takes for the second machine to malfunction and require maintenance. Q is the machine obstruction process (AC) of the second process in the self-produced workshop, which means that when the total processing time of the second process reaches Q , the machine needs to be repaired. quo ( ⋅ ) is taken as an integer. The formula for processing workpieces in only one workshop is shown in the following equation:

The value range constraint for the first processing step is shown in the following equation:

The value range constraint for the second processing step is shown in the following equation:

In Eq. (11), T i is the time when the second working program starts working. Therefore, the PLSM model based on AC and HR has been constructed.

3.2 Design of a hybrid algorithm integrating HR and JR

Due to the fact that the PLSM model based on AC and HR belongs to both the sequence arrangement problem and NP-hard problem in the ALW, a reasonable arrangement of the ALW operation steps and the determination of the optimal solution can greatly improve processing efficiency. Given this, this study uses JR and HR to solve and optimize it. JR is an optimization method used to solve ALW problems, which has the advantage of effectively arranging the construction sequence of projects and optimizing the utilization of resources. It is widely used in construction organization design and production operation planning. If the arrangement of JR satisfies the rule, the following equation is obtained:

In Eq. (12), r represents the processing time of two processes, k represents the workpiece, and h represents the workpiece. At this point, workpiece k is processed before workpiece h . The specific steps of the JR sorting method are illustrated in Figure 4 [19].

Figure 4

Johnson rule example.

In Figure 4, a process matrix for a workpiece is first established, and workpiece 2 is selected and ranked first according to the principle of minimizing the process. Secondly, from Workpiece 4 and Workpiece 5, Workpiece 5 is selected to be placed behind Workpiece 2, and Workpiece 6, which has a smaller process than Workpiece 5, is selected to be placed behind Workpiece 5. Next, according to the above rules, workpiece 1, workpiece 6, and workpiece 3 are sorted to obtain a new process matrix. Finally, based on the maximum process time, the production schedule for workpieces 1 to 6 is calculated. Among them, the right side of the slash in the production schedule represents the time flow of the end of the process, as expressed in the following equation:

In Eq. (13), H is the process time of the end time of the process. W and A are the start time and processing time of this process. In the production process, the start time of a process is determined by two factors. One is the end time of the previous process before this process. The second is the processing end time of the workpiece before the equipment used in this process is tightened. The starting time of this process should be the larger of the two numbers mentioned above. For example, the calculation of process time at the end of workpiece 1 and workpiece 2 is shown in the following equation:

In Eq. (14), E is the processing end time of the workpiece immediately before. ι 1 and τ 2 are the process times at the end of workpiece 1 and workpiece 2. However, although JR can find the optimal solution, there are multiple optimal solutions [20]. HR is a heuristic-based decision-making or problem-solving approach that utilizes simplified rules or experience to quickly find solutions to problems. It has advantages such as speed and practicality [21]. Therefore, this study combines HR and JR to construct a hybrid algorithm based on JR and HR, as shown in Figure 5.

Figure 5

Hybrid algorithm based on JR and HR.

In Figure 5, O I and O o are sets of workpieces processed in the in-house workshop and outsourced workshop. The proposed algorithm flow is as follows. The first step sets O I and O o as the set of artifacts to be processed by the asset shop and the outsourcing shop, respectively. At the same time, the set and decision variables are set to empty sets and 0 values. The second step is to take the absolute value of the difference between the first and second process processing time of all the workpieces to be processed, and sort the array A from the largest to the smallest. In the third step, O I and O o are initialized, resulting in O I = A and O o = ∅ . In the fourth step, the JR is used to sort the processing order of all workpieces, and the minimum manufacturing period of the self-produced workshop and the outsourcing workshop is calculated. Then, the two are subtracted and the absolute value is taken. The fifth step is to take out the workpiece to be processed from A , put it in O o , and set O o = O o − δ i . If A = ∅ is set, then the machining workpiece in the O I and O o are sets and the optimal production sequence in the set are output. The corresponding objective function value is calculated by the output value. Instead, step 6 uses JR to sort the set of machining parts in O I and O I , respectively. The minimum manufacturing period of the two sets is calculated, getting the minimum manufacturing period of the two sets. The difference operation is carried out and the absolute value is taken and denoted as B . If the absolute value is less than B , it is updated to the new value. If the workpiece is processed in an outsourced workshop, combined with the sorting results of the previous step, the start time of the workpiece in the first and second processes can be obtained, and then proceed to the fifth step. If the workpiece is processed in the self-produced workshop, proceeding to the next step. Step 7, O I ∪ δ i and O o = O o − δ i are set to enter the next step. Finally, the workpiece in O I and O o is set and the optimal production sequence in the set are output. The corresponding objective function values are calculated by the output values.

In the process of processing time and cost, the proposed heuristic algorithm takes into account that the second process is the bottleneck process of the problem under study. The processing time of the second process is longer than that of the first process, thereby improving the quality. Therefore, the heuristic algorithm proposed in this paper outsources the workpiece with the biggest difference between the processing time of the second process and the processing time of the first process. This increases the continuity in the asset shop floor and prevents the idle machine time from being wasted. After the workpieces are assigned, the workpieces in each workshop are sorted according to JR. The heuristic algorithm proposed in this paper improves the utilization rate of idle machine tools and reduces the processing cost from the point of view of a minimum manufacturing cycle.

4.1 Performance comparison test of hybrid algorithms

To verify the superiority of the proposed hybrid algorithm (Algorithm 1), its suitability for optimizing production scheduling management is first examined. Then, it is compared experimentally with SHPSO (Algorithm 2), QLINSGA-II (Algorithm 3), and Q-Learning-Sarsa-K-mes-GA (Algorithm 4) in Matlab simulation software. Experimental indicators include changes in optimal solutions and average values. Table 1 shows the experimental parameters of this study.

In the above environment, the algorithm’s performance is first validated using the index i n . It is the ratio of the difference between the minimum manufacturing period with and without outsourcing to the difference between the total cost with and without outsourcing at different scales. In this paper, MATLAB tool is used to verify the effectiveness and functional standard of the designed heuristic algorithm through the target value results of the three rules. Rule 1 (R ₁) states that the greater the total processing time, the higher the priority for outsourcing. Rule 2 (R ₂) compares the processing time of the second process (bottleneck process), and those with longer processing time are given priority for outsourcing. Rule 3 (R ₃) calculates the sum of the processing time for two processes of the workpiece to be processed. The smaller the sum of processing time, the more priority is given to outsourcing. The experimental results show that the algorithm conforms to the following three rules, and the algorithm is effective. The specific contents are shown in Table 2.

In Table 2, R and R 1 are outsourced for each workpiece with a larger difference in processing time and a larger sum of time between the two processes. * indicate the optimal results under different production scales. R 2 and R 3 prioritize outsourcing when the maintenance time for the second process is longer and prioritize outsourcing when the sum of the processing time for the two processes is shorter. Under different production range conditions, the values obtained by this algorithm all comply with R regulations and have better performance. This indicates that overall, as the production scale increases, the algorithm performance becomes better and is suitable for production scheduling management with outsourcing situations. To further verify the superior performance of the algorithm, this study conducts comparative experiments on the optimal solution and average value of four algorithms. The optimal solutions and average results of each algorithm are shown in Figure 6.

Figure 6

The optimal solution of change and the average. (a) The optimal solution variation diagram of each algorithm. (b) Variation diagram of the mean value of each algorithm.

In Figure 6(a), the optimization rates of algorithms 1–4 are 96.7, 90, 78.6, and 84.7%. In Figure 6(b), the average values of algorithms 1–4 are 5998.7, 6287.3, 6698.9, and 6986.8 h. Among them, Algorithm 1 has the lowest average processing time. This indicates that, from the perspective of optimal solution and average value, the performance of the research algorithm is superior to that of the comparison algorithm. The accuracy and recall results of each algorithm are shown in Figure 7.

Figure 7

Comparison results of recall and accuracy of each algorithm. (a) Comparison results of accuracy of each algorithm. (b) Precision comparison results of each algorithm.

In Figure 7(a), Algorithm 1 has the highest average accuracy, at 97.8%, which is higher than Algorithm 2’s 92.6%, Algorithm 3’s 79.9%, and Algorithm 4’s 84.6%. In Figure 7(b), the average recall rates of algorithms 1–4 are 98.8, 97.6, 96.6, and 95.4%. Algorithm 1 has the highest average recall rate. This indicates that Algorithm 1 has better recall and accuracy than the comparison algorithms. Based on the above results, the research algorithm performs the best and is effective in terms of index i n , optimal solution, average, accuracy, and recall at different scales. To further validate the performance of the proposed algorithm (Algorithm 1), it is compared with SHPSO (Algorithm 2), QLINSGA-II (Algorithm 3), and Q-Learning-Sarsa-K-means-GA (Algorithm 4) in the dual resource workshop scheduling dataset. Using F1 value and recall rate as indicators, a performance comparison is made in the ob workshop scheduling problem based on the accuracy and ROC of the JSSP dataset. The comparison results are shown in Table 3.

In Table 3, F1 value, recall, accuracy, and AUC value of the proposed algorithm are 0.98, 0.97, 0.98 and 0.97, respectively, which are superior to the comparative algorithms. The results show that the proposed algorithm is superior in these four dimensions. Having high F1 value, recall, accuracy, and AUC value is an important basis for evaluating the robustness of the algorithm model. Therefore, the above results show that the proposed model is robust.

4.2 Performance analysis of engineering PLSM model

To verify the performance superiority of the proposed engineering PLSM model, an analysis is conducted on the scheduling scheme with and without outsourcing and the corresponding algorithm’s target values. In this experiment, Q = 10 , Γ = 10 , μ = 20 , w = 0.6 , p 2 / 0.9 = p 1 , q 2 / 0.9 = q 1 , m = 0.6 , α = − 1 , and the p 1 has values ranging from 1 to 6. The values are randomly generated according to the equal distribution, and the range of p 2 values is between 5 and 10. The scheduling plan and effect of this workpiece with and without outsourcing are shown in Figure 8.

Figure 8

The scheduling scheme and effect of a random group of workpieces with and without outsourcing. (a) Scheduling optimization without outsourcing. (b) There are outsourcing scheduling optimization schemes.

In Figure 8, regardless of whether there is outsourcing, the self-produced workshop experienced a malfunction during the second processing step and required equipment maintenance. The total cost without outsourcing and with outsourcing ( i n = 0.7348) is 269 yuan and 292.68 yuan, respectively. The minimum manufacturing time is 50.8 and 33.4 h. The objective function values are 137.68 and 137.17. This indicates that production scheduling with outsourcing has a faster processing speed, takes less time, and has lower costs when the w time target parameter is greater than 0.6. During the production process, for every unit of cost consumed, 0.7348 working hours can be saved, and utilizing outsourcing to optimize scheduling management has a significant effect. To better compare the results of scheduling optimization, this study introduces a function η = [1 − (the ratio of the objective function values with and without outsourcing scheduling schemes)] × 100%. p 2 = β p 1 and q 2 = β q 1 , where p 1 and q 1 randomly generate values from the [3,10] and [ 3 α , 20 α ] intervals according to a uniform distribution. The comparative results of the effects of α and θ on outsourcing performance when β = 0.9 and β = 1 are shown in Figure 9.

Figure 9

When β = 0.9 and β = 1, α and θ affect the comparison results of the outsourcing effect. (a) Comparison of the influence of α and θ on outsourcing effect β = 0.9. (b) Comparison of the influence of α and θ on outsourcing effect when β = 1.

In Figure 9, when β = 1 increases, as θ increases, the value of η decreases from 1.78 to 0.45%, until −0.83%, and the value of i n becomes smaller and smaller. Whether it is β = 1 or β = 0.9 , as the α value increases, the value of η increases from 1.78 to 3.1%, and the i n value increases from 0.372 to 0.478. As the value of β becomes smaller, both the value of η and i n are higher than the value at β = 1 . The above results indicate that the longer the processing time required for the second process and the higher the production rate in the outsourcing workshop, the more significant the outsourcing effect, which is in line with the actual market demand. To further validate the superiority of the research model (Model 1), it is compared with models based on SHPSO (Model 2), QLINSGA-II (Model 3), and Q-Learning-Sarsa-K-mes-GA (Model 4). The experimental indicators are processing time and processing cost. The comparison results of processing time and processing cost for each model are shown in Figure 10.

Figure 10

Comparison results of processing time and processing cost of each model. (a) Comparison of the processing time of each model. (b) Comparison of processing cost of each model.

In Figure 10(a), the average processing time of models 1–4 is 196.7, 396.8, 226.7, and 498.2 h, with model 1 having the shortest processing time. In Figure 10(b), the average processing cost of Model 1 is the lowest, at 1456.7 yuan, lower than Model 2’s 3897.4 yuan, Model 3’s 2346.1 yuan, and Model 4’s 4968.6 yuan. This indicates that Model 1 has better processing time and cost advantages. The above results indicate that the proposed engineering PLSM model has good performance in terms of outsourcing scheduling, different parameter influences, processing time, and processing cost dimensions. To verify the application effect of the model proposed in the study, it is combined with Models 2–4 to perform optimization scheduling experiments on 3,000 motor shafts processed in a factory. The specific results are shown in Table 4.

In Table 4, under the same conditions, the processing cost, processing time, and machine tool utilization rate of model 1 are 395,102 yuan, 267.2 h, and 98.70%, respectively, which are superior to the comparison model. The above results show that the proposed model is practical from the dimensions of machining cost, machining time, and machine tool utilization.